In this paper we consider the orienteering problem in undirected and directed graphs and obtain improved approximation algorithms. The point to point-orienteering-problem is the following: Given an edgeweighted graph G = (V, E) (directed or undirected), two nodes s, t ∈ V and a budget B, find an s-t walk in G of total length at most B that maximizes the number of distinct nodes visited by the walk. This problem is closely related to tour problems such as TSP as well as network design problems such as k-MST. Our main results are the following. • A 2 + ɛ approximation in undirected graphs, improving upon the 3-approximation from [5]. • An O(log 2 OPT) approximation in directed graphs. Previously, only a quasi-polynomial time algorithm achieved a poly-logarithmic approximation [12] (a ratio of O(log OPT)). The above results are based on, or lead to, improved algorithms for several other related problems.

Mobile sensors cover more area over a period of time than the same number of stationary sensors. However, the quality of coverage achieved by mobile sensors depends on the velocity, mobility pattern, number of mobile sensors deployed and the dynamics of the phenomenon being sensed. The gains attained by mobile sensors over static sensors and the optimal motion strategies for mobile sensors are not well understood. In this paper we consider the problem of event capture using mobile sensors. The events of interest arrive at certain points in the sensor field and fade away according to arrival and departure time distributions. An event is said to be captured if it is sensed by one of the mobile sensors before it fades away. For this scenario we analyze how the quality of coverage scales with the velocity, path and number of mobile sensors. We characterize the cases where the deployment of mobile sensors has

We present a fast algorithm for computing all shortest paths between source nodes s ∈ S and target nodes t ∈ T. This problem is important as an initial step for many operations research problems (e.g., the vehicle routing problem), which require the distances between S and T as input. Our approach is based on highway hierarchies, which are also used for the currently fastest speedup techniques for shortest path queries in road networks. We show how to use highway hierarchies so that for example, a 10 000 × 10 000 distance table in the European road network can be computed in about one minute. These results are based on a simple basic idea, several refinements, and careful engineering of the approach. We also explain how the approach can be parallelized and how the computation can be restricted to computing only the k closest connections.

We provide an overview of an emerging area of domination analysis (DA) of combinatorial optimization algorithms and problems. We consider DA theory and its relevance to computational practice. 1

Abstract. The Prize-Collecting Steiner Tree Problem (PCST) on a graph with edge costs and vertex profits asks for a subtree minimizing the sum of the total cost of all edges in the subtree plus the total profit of all vertices not contained in the subtree. PCST appears frequently in the design of utility networks where profit generating customers and the network connecting them have to be chosen in the most profitable way. Our main contribution is the formulation and implementation of a branch-and-cut algorithm based on a directed graph model where we combine several state-of-the-art methods previously used for the Steiner tree problem. Our method outperforms the previously published results on the standard benchmark set of problems. We can solve all benchmark instances from the literature to optimality, including some of them for which the optimum was not known. Compared to a recent algorithm by Lucena and Resende, our new method is faster by more than two orders of magnitude. We also introduce a new class of more challenging instances and present computational results for them. Finally, for a set of large-scale real-world instances arising in the design of fiber optic networks, we also obtain optimal solution values. Keywords: Branch-and-Cut – Steiner Arborescence – Prize Collecting – Network Design 1

Abstract. We consider two on-line versions of the asymmetric traveling salesman problem with triangle inequality. For the homing version, in which the salesman is required to return in the city where it started from, we give a 3+√5-competitive algorithm and prove that this is best 2 possible. For the nomadic version, the on-line analogue of the shortest asymmetric hamiltonian path problem, we show that the competitive ratio of any on-line algorithm depends on the amount of asymmetry of the space in which the salesman moves. We also give bounds on the competitive ratio of on-line algorithms that are zealous, that is, in which the salesman cannot stay idle when some city can be served. 1

In recent years, there has been much interest in phase transitions of combinatorial problems.

Given a matrix of values in which the rows correspond to objects and the columns correspond to features of the objects, rearrangement clustering is the problem of rearranging the rows of the matrix such that the sum of the similarities between adjacent rows is maximized. Referred to by various names and reinvented several times, this clustering technique has been extensively used in many fields over the last three decades. In this paper, we point out two critical pitfalls that have been previously overlooked. The first pitfall is deleterious when rearrangement clustering is applied to objects that form natural clusters. The second concerns a similarity metric that is commonly used. We present an algorithm that overcomes these pitfalls. This algorithm is based on a variation of the Traveling Salesman Problem. It offers an extra benefit as it automatically determines cluster boundaries. Using this algorithm, we optimally solve four benchmark problems and a 2,467-gene expression data clustering problem. As expected, our new algorithm identifies better clusters than those found by previous approaches in all five cases. Overall, our results demonstrate the benefits of rectifying the pitfalls and exemplify the usefulness of this clustering technique. Our code is available at our websites.

The conventional approach to solving optimization and search problems is to apply a variety of search algorithms to the problem at hand, in order to discover a technique that is well-adapted to the search space being explored. This paper investigates an alternative approach, in which search algorithms are automatically synthesized for particular optimization problem instances. A language composed of potentially useful basic search primitives is derived. This search language is then used with genetic programming to derive search algorithms. The genetic programming system evaluates the fitness of each search algorithm by applying it to a binary-encoded optimization problem (Traveling Salesman), and measuring the relative performance of that algorithm in finding a solution to the problem. It is shown that the evolved search algorithms often display consistent characteristics with respect to the corresponding problem instance to which they are applied. For example, some problem instances are best suited to hill-climbing, while others are better adapted to conventional genetic algorithms. As is to be expected, the search algorithm derived is strongly dependent the scale and representation of the problem explored, the amount of computational effort allotted to the overall search, and the search primitives available for the algorithm. Additionally, some insights are gained into issues of genetic algorithm search. A novel “memetic crossover ” operator was evolved during the course of this research.

Abstract. In multi-criteria optimization, several objective functions are to be optimized. Since the different objective functions are usually in conflict with each other, one cannot consider only one particular solution as optimal. Instead, the aim is to compute so-called Pareto curves. Since Pareto curves cannot be computed efficiently in general, we have to be content with approximations to them. We are concerned with approximating Pareto curves of multi-criteria traveling salesman problems (TSP). We provide algorithms for computing approximate Pareto curves for the symmetric TSP with triangle inequality ( ∆-STSP), symmetric and asymmetric TSP with strengthened triangle inequality (∆(γ)-STSP and ∆(γ)-ATSP), and symmetric and asymmetric TSP with weights one and two (STSP(1, 2) and ATSP(1, 2)). We design a deterministic polynomial-time algorithm that computes (1 + γ + ε)-approximate Pareto curves for multi-criteria ∆(γ)-STSP for γ ∈ [ 1, 1]. We also present two randomized approximation algo-2 rithms for multi-criteria ∆(γ)-STSP achieving approximation ratios of